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phyky
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if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?
phyky said:if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?
Nugatory said:We can compute both, they just won't be equal.
Ravi Mohan said:In some cases (when the commutation is a projector to a particular eigen space) they might be equal.
Nugatory said:We can compute both, they just won't be equal.
Then the expectation is zero. Both operators map from Hilbert Space to Hilbert Space in this context. And you can always take an inner product between two vectors in Hilbert Space. They might be orthogonal, because they belong to different sub-spaces, but then the inner product is trivially zero and that's your expectation value.dextercioby said:Generally we can't, because if the vector psi is in Dom(AB), it may not be in Dom(BA).
phyky said:conclusion is if AB is non commute. we can only compute 1 of the expectation value, either <|AB|> or <|BA|>?
The expectation value for a non-commuting operator is a mathematical concept used in quantum mechanics to calculate the average value of a physical quantity measured in a quantum system. It is represented by the notation ⟨A⟩, where A is the operator and the brackets indicate the expectation value.
The calculation of the expectation value for non-commuting operators involves finding the eigenstates of the operators and using the corresponding eigenvalues in a weighted average formula. The formula is given by ⟨A⟩ = ∑n Pn an, where Pn is the probability of measuring the eigenvalue an in the state ⟨n⟩.
The expectation value for non-commuting operators is an important concept in quantum mechanics as it allows us to make predictions about the behavior of a quantum system. It is also used to determine the uncertainty in the measurement of a physical quantity, as given by the Heisenberg's uncertainty principle.
Yes, the expectation value for non-commuting operators can be negative. It is possible for the eigenvalues of the operator to have both positive and negative values, which will affect the overall calculation of the expectation value. However, the expectation value itself is not a physical quantity and does not have a direct physical interpretation.
The Heisenberg's uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This uncertainty is mathematically represented by the non-commutativity of certain operators, which in turn affects the calculation of the expectation value. Therefore, the concept of expectation value for non-commuting operators is closely related to the uncertainty principle in quantum mechanics.